Rivasseau, Vincent | Engineering Papers

DOE OSTI2015

Corrected loop vertex expansion for Φ{sub 2}{sup 4} theory

This paper is an extended erratum to Rivasseau and Wang [J. Math. Phys. 53, 042302 (2012); e-print http://arxiv.org/abs/1104.3443 [math-ph]], in which the classic construction and Borel summability of the ϕ{sub 2}{sup 4} Euclidean quantum field theory was revisited combining a multi-scale analysis with the constructive method called Loop Vertex Expansion (LVE). Unfortunately we discovered an important error in the method of Rivasseau and Wang [J. Math. Phys. 53, 042302 (2012); e-print http://arxiv.org/abs/1104.3443 [math-ph]]. We explain the mistake, and provide a new, correct construction of the ϕ{sub 2}{sup 4} theory according to the LVE.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2014

Generalized constructive tree weights

The Loop Vertex Expansion (LVE) is a quantum field theory (QFT) method which explicitly computes the Borel sum of Feynman perturbation series. This LVE relies in a crucial way on symmetric tree weights which define a measure on the set of spanning trees of any connected graph. In this paper we generalize this method by defining new tree weights. They depend on the choice of a partition of a set of vertices of the graph, and when the partition is non-trivial, they are no longer symmetric under permutation of vertices. Nevertheless we prove they have the required positivity property to lead to a convergent LVE; in fact we formulate this positivity property precisely for the first time. Our generalized tree weights are inspired by the Brydges-Battle-Federbush work on cluster expansions and could be particularly suited to the computation of connected functions in QFT. Several concrete examples are explicitly given.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗